I have seen this faq around, and its mostly correct, but here's some edits which I think are needed: (banned from Isonews so cannot edit on there)

Firstly, there were chips for Japanese PS2 consoles that allowed PSX games to play, then the following became available:

NEO 1 - originating from the UK, this was the first PAL supporting PS2 modchip. NEO1 was designed for PSX backups on a PS2, and it was later found to be capable of booting PS2 games using an AR2 from Datel by static, a well known ISO group. The chip unfortunately could not play PSX imports (backups or original), and did not support EA PS2 games, but PSX import backups would play when patched to the correct format. The chip also required an additional small value capacitor soldered to improve reliability.

NEO 2 - PS2 modchip designed for PS2 backups to be played with a swap with GS2/AR2 and PSX backups without a swap. This mod used 8 wires and was very popular, unfortunately it was not capable of playing imported PSX games (originals or backups), but import backups would play when patched to the correct format. The embedded hex was ripped from a de-capped chip and released onto the internet, thus this became the widest used modchip in PS2 consoles in the world. The chip also required an additional small value capacitor soldered to improve booting reliability.

NEO 2 pro- PS2 modchip designed for PS2 backups to be played with a swap with GS2/AR2 and PSX backups without a swap. This mod used 8 wires and was very popular, unfortunately it was not capable of playing imported PSX games (originals or backups), but import backups would play when patched to the correct format. The NEO 2 pro was PCB mounted, and the small value capacitor was on-board the PCB.

NEO 2.2 - NEO2.2 is an updated version of NEO2, it allowed more time for EA boot, and was also capable of stopping the spin of the disks for use with AR2 / GS2 versions that did not stop the disk spin automatically for use with booting EA PS2 games. This mod required 9 wires to be soldered to the motherboard for full functionality. The chip also required an additional small value capacitor soldered to improve reliability in some cases, timing improvements had been made to this version, but was not completely successful hence some requirement for the capacitor. This NEO was also available in the 'pro' PCB version.

STARCHIP - Originating from the USA, it has similar features to Neo2.2, and was the first to utilize the disk-stop feature for EA games when swapping with AR2 or GS2 that did not support the disk-stop feature. This mod required 8 wires to be soldered to the motherboard for full functionality, its booting capability was more reliable than the NEO version of chips as attention had been given to timing problems, although a small value capacitor was still required to be soldered in many cases to improve its booting efficiency.

NEOKEY - Originating from Asia, the first external mod for PS2. The NEOKey still requires soldering to the PS2's mainboard, therefore many people are 'illusioned' by it being called an external modchip. The NeoKey requires one swap to be performed with a GS2 / AR2 for PS2 backups, no swap for PSX backups. The NeoKey does not support the EA swap. The NEOKey also is known as the USBMod and SBox. This mod requires 1 wire (2 points) to be soldered to the motherboard for functionality. Unfortunately the NEO KEY cannot play imported PSX games (originals or backups), although backup imported PSX games will play when patched. The NEO Key and its clones are small modules that plug into the external USB connector of the PS2.

NEO 3/BOOTMASTER 3 - Originating from the UK, this was the first modchip that supported all the usual booting features, but with additional PSX import games, disk-stop and DVD multiregion with Macrovision disable features on the PS2 in one single design. Initially it was an un-named project by Channeltech for themselves, and negotiation with NEO as being the sole distributor gave it the name as NEO 3. NEO lost interest in the project due to delays and the beginning of development of the NEO 4 so Channeltech renamed the project to Bootmaster 3 and completed and released the design. The Bootmaster 3 was a 3-chip and multi-component PCB deisgn.

ORIGA 1.X - Originating from Italy, the Origa will boot PS2 backups with one swap using AR2/GS2, DVDR or CDR. The mod will also boot IMPORT ORIGINAL PS2 games with no swap. This mod requires 17 wires to be soldered to the motherboard. This chip unfortunately had some small bugs, one which caused the Playstation 2 logo to be distorted on boot, and awkward booting methods.

ORIGA 2.0 - This is currently in development by Origa brothers, it is known that they are making enquiries into FBGA purchase and are seeking to release a new design. This design is also known as the NSF chip (non swap final), although Origa may decide to give the mod a new name should it be released.

NOVACHIP - Origa 1.0 clone with less wires (from the same camp). This mod requires 14 wires to be soldered to the motherboard for functionality. The chip is PCB mounted.

NEO 2.5 / XKEY - Swap based mod with same features as Neo2.2 and less wires to install. You have to hold reset for every game you play. Only available for V4 and V5 PS2s. This mod requires 5 wires to be soldered to the motherboard for functionality.

NEO 2.6 - Same as Neo2.5, but automatically chooses the GS2/AR2 selections. This mod requires 6 wires to be soldered to the motherboard for 100% functionality.

NEO4 - Originating from the UK, this was the first non-swap CDR mod. The Neo4 will boot PS2 CDR games with no swap directly. PS2 DVDRs still require one swap with a larger TOC original DVD game. This mod also provides multi-region DVD and Macrovision disable capability, and the ability to boot Original and backup import PSX games. Unfortunately the mod does not boot PS2 EA games directly, including originals which means it is also the first chip released that stops original games from booting (only EA games). The mod also does not support original import PS2 game booting. The mod is a multi-chip PCB design and requires 44 wires to install to full functionality. There has been some vendors offering this chip installed along with an Origa version of chip to allow direct boot of DVDR and import original PS2 games, but the wiring count becomes very high in the console, and booting reliability is poor.

NEO4.1 - Updated verion of Neo4. The Neo4.1 supports more consoles, has the latest DVD drivers and faster boot than the Neo4, This mod requires 44 wires to be soldered to the motherboard for 100% functionality, although a version that used a plastic 'clip' for connecting to the bios chip and reducing the number of wires was released, but the clip has proven to be very unreliable. There has been some vendors offering this chip installed along with an Origa version of chip to allow direct boot of DVDR and import original PS2 games, but the wiring count becomes very high in the console, and booting reliability is poor.

NEOSTAR - Originating from easybuy2000 in the USA, this was a fabled Neo4 clone that did not exist.

MESSIAH - Originating from the UK, this modchip supports all game booting functions, all PS2 CDR / DVDR and PSX CDR will boot, and all import versions of game will boot, everything directly without swap, and also at full screen playback. The mod supports all PAL consoles, and one version of V4 USA console (gap bios). The Messiah is a single FBGA chip PCB mounted design and requires 23 wires to be connected for functionality, it is available in many different names (eg. Elvis, Goliath, Phoenix etc. etc).

R-MOD - The mod that isnt a mod ?. The R-Mod imitates the Knife Trick and makes it unable to play all games, a very limited performance design.

Hey CT thanks for correcting me. Seems like all the big names are correcting me (Raptor also helped). I have also got an update coming (I just finished it last night, but I decided to proofread and it was very late, so I decided to wait) but it will have mainly Gazza's info and info on future Neo chips, supplied to me by Raptor.